Formula for Angle between Two Lines

Here, you will learn formula for angle between two lines, equation of straight line making an angle with a given line and reflection and image of a point in a line and also length of perpendicular from a point on a line.

Let’s begin –

Angle between Two Lines

(a)  If \(\theta\) be the angle between two lines : y = \(m_1x + c_1\) and y = \(m_2x + c_2\), then

tan\(\theta\) = \(\pm\) (\(m_1-m_2\over {1+m_1m_2}\))

(b)  If the equation of lines are \(a_1x + b_1y + c_1\) = 0 and \(a_2x + b_2y + c_2\) = 0, then these lines are – 

(i)  Parallel  \(\iff\)   \(a_1\over a_2\) = \(b_1\over b_2\) \(\ne\) \(c_1\over c_2\)

(ii)  Perpendicular  \(\iff\)  \(a_1a_2\) + \(b_1b_2\) = 0

(iii)  Coincident  \(\iff\)  \(a_1\over a_2\) = \(b_1\over b_2\) = \(c_1\over c_2\)

Equation of Straight line making an angle with a Line :

Equation of line passing through a point (\(x_1,y_1\)) and making an angle \(\alpha\), with the line y = mx + c is written as

y – \(y_1\) = \(m \pm tan\alpha\over {1 \mp mtan\alpha}\)(x – \(x_1\))

Reflection and image of a point in a line :

Let P(x,y) be any point, then its image with respect to

(a)  x-axis is Q(x,-y)

(b)  y-axis is R(-x, y)

(c)  origin is S(-x,-y)

(d)  line y = x is T(y,x)

(e)  Reflection of a point about any arbitrary line : The image (h,k) of a point P(\(x_1,y_1\)) about the line ax+by+c = 0 is given by following formula.

\(h-x_1\over a\) = \(k-y_1\over b\) = -2(\(ax_1+by_1+c\over {a^2+b^2}\))

and the foot of perpendicular (p,q) from a point (\(x_1,y_1\)) on the line ax+by+c = 0 is given by following formula.

\(p-x_1\over a\) = \(q-y_1\over b\) = -(\(ax_1+by_1+c\over {a^2+b^2}\))

Length of perpendicular from a point on a line :

Length of perpendicular from a point (\(x_1,y_1\)) on the line ax + by + c = 0 is

|\(ax_1 + by_1 + c\over {\sqrt{a^2+b^2}}\)|

In particular, the length of perpendicular from the origin on the line ax + by + c = 0 is

P = \(|c|\over {\sqrt{a^2+b^2}}\).

Hope you learnt formula for angle between two lines and all other concepts. Practice more questions to learn more and get ahead in competition. Good Luck!

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